How to solve Systems of Equations with Three Variables | Elimination method

[00:00:00]Hello everybody. Welcome to this platform.

[00:00:04]This is Sijamba Jacob and I'm pretty sure each and everybody watching the video right now is doing good.

[00:00:11]We've got these systems of equation right over here. Or should I say equations? There are three of them which we need to solve.

[00:00:20]But before we solve this, I want you to just get to have an idea of what you need to do when you've been given something that looks like this.

[00:00:29]These are simultaneous uh equations.

[00:00:32]So how are we supposed to solve this? We can solve by elimination.

[00:00:36]We first put a bar and then get the coefficient put it up there and then this other coefficient you write it down here.

[00:00:44]You change the sign, okay? If uh the coefficients that you've picked this side, they've got similar uh signs or should I say if they are both positives or both negatives one of them should have a different sign. So for example here, it was positive 11 and then positive three. So we just introduce a negative on one of them so that they're different.

[00:01:12]And from this point, you now start multiplying whatever is here by everything. So here, we're getting in uh positive it's positive 55 y, okay? So if we don't write we know that it's positive because -11 * uh 5y will give us uh a positive 55.

[00:01:35]And from here, when we multiply, we're getting a negative 33 z which is equal to -121 when you multiply.

[00:01:46]And from this point, we now multiply what is down here. Here it's giving us a -39 y okay?

[00:01:56]And then here, positive 33 z which is equal to 105 when you multiply.

[00:02:04]And when you reach this point you just have to add everything. So, you add like this.

[00:02:14]So, here we're getting a 16 y.

[00:02:17]While these two cancel and here we're getting a negative 16.

[00:02:22]So, the idea of uh putting the coefficients here, we want to come and cancel one set of variables.

[00:02:31]So, the letters, for example, here the z have canceled. We've remained with the the y.

[00:02:37]And you can divide here here the value of y is coming out to be equal to negative one because uh um Oh, this is this is positive.

[00:02:50]This is positive. So, this positive and negative will give will give us a negative.

[00:02:54]16 into itself it's a one. Into 16 it's also a one.

[00:02:58]So, it has given us y to be equal to negative uh negative one.

[00:03:04]So, once you get to find this you can now just say, "Okay. For me to find the value of z we just get any of the equations here.

[00:03:13]I'll get the first one.

[00:03:16]And then after getting the first one you substitute the value of y.

[00:03:22]We we know, of course, that we found the value of y to be equal to negative one.

[00:03:28]So, you just say negative five negative one plus three z is equal to 11.

[00:03:35]Here we're getting a five plus three z is equal to 11.

[00:03:40]We subtract.

[00:03:42]This will give us a three z to be equal to six. Divide by three by three. This is z.

[00:03:50]So, the value of z is coming out to be equal to a two.

[00:03:55]So, here these are the solutions and this is what we call elimination method. Now, what about if you meet up something that looks like this?

[00:04:05]What is it that you're supposed to do?

[00:04:08]Without wasting your time, let's jump into it. The first thing should be to label.

[00:04:17]Okay, so indicate because there are three equations.

[00:04:20]This will help us be able to know what we need to do.

[00:04:25]So, I'll get one of the equations. In my case, I'll get equation two.

[00:04:31]And um for this equation two, you make one of the variables the subject.

[00:04:40]So, you can see all the terms on equation two have gone to the other side. I've just remained with X.

[00:04:48]That's what you need to know.

[00:04:49]You make one of the variables the subject, which I've done right over here.

[00:04:57]And after that, you call this as equation number four.

[00:05:01]They are now four.

[00:05:04]So, in equation one, there is X there.

[00:05:08]Now, in place of X, you're going to substitute this.

[00:05:16]All this part will be where X is. So, it will be two open bracket -6 -3y + 2z + y + y -z is equal to -1.

[00:05:33]That's what you need to do.

[00:05:36]Let me clean up.

[00:05:40]So, after this, you you simply simplify.

[00:05:49]So, you're getting -12 minus uh 6y plus 4 z plus y minus z is equal to negative uh negative one.

[00:06:06]So, you can see you just multiply.

[00:06:10]And this is what you're getting.

[00:06:12]So, from this point you can group the like terms. So, this and this we're getting a -5y.

[00:06:21]While this and this we're getting a plus uh 3z which is equal to -1 + 12.

[00:06:28]So, -5y + 3z is equal to 11. So, this is equation uh equation five.

[00:06:38]So, we've now formed five equations.

[00:06:42]We go back to this point. This time around we substitute right here. So, remember equation two is the key which helped us to come up with equation four.

[00:06:55]Whatever we have on equation four should be substituted on equation one and then we simplify and it will look like this.

[00:07:02]And also the same should be substituted in equation five.

[00:07:06]So, in equation five I mean equation three it will look like this when we substitute.

[00:07:14]So, it's a matter of substituting.

[00:07:21]Okay. So, here we've gotten this part. Put it where x is because we're saying this is equal to x. And this is how it will look like.

[00:07:32]When we simplify, this is giving us a -30 15y plus 10z and then I've got 2y + z is equal to 5.

[00:07:44]So, um these two terms are the same.

[00:07:50]I think here it's supposed to be negative, not positive. They're the same, so get -13y.

[00:07:58]And then this is z.

[00:08:00]So, these two terms are the same, they'll give us a 11 11z, which is equal to 5 + 30. So, this -30 it will be positive 30.

[00:08:14]And from here we've got -13y + 11z is equal to 35.

[00:08:21]So, this is equation six.

[00:08:24]So, now now that we've reached this point, we can just get these two equations and solve them simultaneously. So, if you're able to see, these are the two equations that we started with.

[00:08:37]Those are the two equations that we started with. Let me write them here.

[00:08:42]We've got uh Okay, -5y + 3z is equal to 11, which is this one. For this one, we've got uh -13 y + 11z, which is equal to 35.

[00:09:03]Yes, 35.

[00:09:06]So, from here we can simply we can simply solve.

[00:09:17]So, according to what we we did, we got the coefficient here and there. We change the sign. This gave us 55y.

[00:09:26]And then here it gave us a -33z, which is equal to -121.

[00:09:32]For this one it gave us a -39y + 33z is equal to 105.

[00:09:40]And then we add it.

[00:09:42]This give us a When we add here it give us a a 16Y. Well, these two cancelled. Here we got a negative 16.

[00:09:53]Where we divided and it give us a Y is equal to -1.

[00:09:59]That's what we did. And from here when we substitute in any of the equation, you know what case we substituted in the first equation.

[00:10:07]That was -5Y plus 3Z is equal to 11 -5.

[00:10:14]Okay? Like this.

[00:10:16]Give us something that looked like this.

[00:10:20]Okay? Here it give us a positive five. So, this is 3Z is equal to 11 minus five. That's how it was.

[00:10:30]So, this is a 3Z is equal to six. We divide by divided by three by three.

[00:10:36]Uh Z was equal to a two, if you're able to remember. So, these are the solutions that we found for Z and Y.

[00:10:45]So, that we've got the solutions for Z and Y we can now find Y using this one. We know that X is equal to -6 -3Y plus 2Z.

[00:11:00]We just substitute.

[00:11:02]The value of Y is a -1.

[00:11:05]Plus two.

[00:11:07]Z is a two.

[00:11:09]This is giving us -6 plus three plus four.

[00:11:13]So, when we simplify here, we're getting something that looks like this and this is giving us a one.

[00:11:18]So, the value of X is a one.

[00:11:21]Thank you so much for watching. Please remember to share the video if you're watching it on YouTube or Facebook or TikTok. Remember to share.

[00:11:31]And also, please subscribe to the YouTube channel. The name is Jacob Sichamba.

[00:11:36]So, in my next video, I'll try to show you how to solve this same system of equation using what we call Cramer's rule. Okay?

[00:11:45]That's the idea of using determinants to find to solve the equation. So, please make sure that you set that notification bell such that whenever I share something here, you are notified. Thank you, and bye-bye.